Internal problem ID [9944]
Internal file name [OUTPUT/8891_Monday_June_06_2022_05_46_13_AM_32391535/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1622 (6.32).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_x"
Maple gives the following as the ode type
[[_2nd_order, _missing_x]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\left (y+3 a \right ) y^{\prime }-y^{3}+a y^{2}+2 a^{2} y=0} \]
This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(y\) an independent variable. Using \begin {align*} y' &= p(y) \end {align*}
Then \begin {align*} y'' &= \frac {dp}{dx}\\ &= \frac {dy}{dx} \frac {dp}{dy}\\ &= p \frac {dp}{dy} \end {align*}
Hence the ode becomes \begin {align*} p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+\left (y +3 a \right ) p \left (y \right )+\left (2 a^{2}+y a -y^{2}\right ) y = 0 \end {align*}
Which is now solved as first order ode for \(p(y)\). Unable to determine ODE type.
Unable to solve. Terminating
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, diff(y(x), x) = -y(x)*a, y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful Try integration with the canonical coordinates of the symmetry [exp(a*x), -exp(a*x)*y*a] -> Calling odsolve with the ODE`, diff(_b(_a), _a) = -_a^3*_b(_a)^3+_a*_b(_a)^2, _b(_a), explicit, HINT = [[_a, -2*_b]]` *** Suble symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[_a, -2*_b]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 416
dsolve(diff(diff(y(x),x),x)+(y(x)+3*a)*diff(y(x),x)-y(x)^3+a*y(x)^2+2*a^2*y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}-\frac {-\textit {\_f}^{8}+c_{1} \textit {\_f}^{2}-{\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {2}{3}}}{\left (-\textit {\_f}^{6}+c_{1} \right ) {\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {1}{3}}}d \textit {\_f} \right ) a +c_{2} a +{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}\frac {-i \sqrt {3}\, \textit {\_f}^{8}+\textit {\_f}^{8}+i \sqrt {3}\, c_{1} \textit {\_f}^{2}+i \sqrt {3}\, {\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {2}{3}}-c_{1} \textit {\_f}^{2}+{\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {2}{3}}}{\left (-\textit {\_f}^{6}+c_{1} \right ) {\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {1}{3}}}d \textit {\_f} \right ) a +2 c_{2} a +2 \,{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\ y \left (x \right ) &= \operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {-i \sqrt {3}\, \textit {\_f}^{8}-\textit {\_f}^{8}+i \sqrt {3}\, c_{1} \textit {\_f}^{2}+i \sqrt {3}\, {\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {2}{3}}+c_{1} \textit {\_f}^{2}-{\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {2}{3}}}{\left (-\textit {\_f}^{6}+c_{1} \right ) {\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {1}{3}}}d \textit {\_f} \right ) a +2 c_{2} a +2 \,{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 58.636 (sec). Leaf size: 185
DSolve[2*a^2*y[x] + a*y[x]^2 - y[x]^3 + (3*a + y[x])*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {c_1 \wp '(x c_1+c_2;0,1)}{\wp (x c_1+c_2;0,1)} & a&=0 \\ -\frac {e^{-a x} c_1 \wp '\left (\frac {e^{-a x} c_1}{a}+c_2;0,1\right )}{\wp \left (\frac {e^{-a x} c_1}{a}+c_2;0,1\right )} & \text {True} \\ \end {array} \\ \end {array} \\ y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} c_1 & a&=0 \\ -e^{-a x} c_1 & \text {True} \\ \end {array} \\ \end {array} \\ y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} -\frac {e^{-a x} c_1 \wp '\left (\frac {e^{-a x} c_1}{a};0,1\right )}{\wp \left (\frac {e^{-a x} c_1}{a};0,1\right )} & a\neq 0 \\ \frac {c_1 \wp '(x c_1;0,1)}{\wp (x c_1;0,1)} & \text {True} \\ \end {array} \\ \end {array} \\ \end{align*}